Question

line t is tangent to the circle. find the measure of arc def. the measure of arc def is \\(\square^\circ\\).

Explanation:

Step1: Find arc DF

The measure of a tangent-chord angle is half the measure of its intercepted arc. So:
$$m\angle tDF = \frac{1}{2}m\overset{\frown}{DF}$$
$$117^\circ = \frac{1}{2}m\overset{\frown}{DF}$$
$$m\overset{\frown}{DF} = 117^\circ \times 2 = 234^\circ$$

Step2: Find arc DEF

A full circle is $360^\circ$. Subtract arc DF from $360^\circ$:
$$m\overset{\frown}{DEF} = 360^\circ - m\overset{\frown}{DF}$$
$$m\overset{\frown}{DEF} = 360^\circ - 234^\circ = 126^\circ$$

Answer:

$126$